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Beyond the mesh: A new method for solving equations

The image on the left shows an RBF-type discretization for modeling in 3D spherical shells. The image on the right uses the discretization in the left image to show an example of 3D thermal convection in a highly convective regime. The red ball denotes the inner boundary, whereas blue denotes downwelling plumes and yellow denotes upwelling plumes. (Image courtesy Natasha Flyer.)

January 11, 2010 | NCAR scientist Natasha Flyer is using an innovative method known as radial basis function (RBF) to model simple physical processes in the geosciences. The research is poised to offer a new way of solving equations that could significantly improve models used by atmospheric and solar scientists.

In the context of physical modeling, RBFs are in the early stage of development, but they show great promise, according to Flyer. Unlike other methods employed in the geosciences, RBFs do not require the use of any meshes. They are naturally defined on scattered nodes in any number of dimensions, combining geometric flexibility with high accuracy. Instead, values are defined by their distance from one another, as opposed to their location in any particular coordinate systems. Because of this mesh-free character, the RBF approach is well suited to the variable resolution that is helpful when modeling atmospheric and solar phenomena.

Flyer has successfully applied RBFs to a number of physical processes: basic tsunami modeling, 3-D convection in Earth's mantle, and non-linear shallow water equations. She sees potential for the method's application to problems involving irregular geometries in multiple dimensions-for example, providing a better understanding of physical processes within the Sun's corona.

"I'm interested in creating highly accurate, yet numerically simple and elegant mathematical methods that will give strong insights into the workings of the Sun-Earth system," says Flyer, who is a mathematician by training. "I look at the basic underlying math of the equations that represent the physical models."

Flyer's work is especially groundbreaking because she is one of only a couple of researchers in the United States who develop RBFs for application to the geosciences. "If this works out, it's going to be an entirely new way to look at equations," she says.